Second-order time integration of the wave equation with dispersion correction procedures

Abstract

Second-order time integration of the wave equation is numerically efficient with time steps close to the limit set by the stability criterion. However, the dispersion errors over realistic propagation distances are unacceptable with time steps in this range. A common procedure is to perform a postpropagation correction on the numerically simulated field. The post-propagation correction does not always provide a sufficiently accurate result. The reason is explained as a lacking source correction term. The proper source correction procedure is identified. Dispersion free results, using secondorder time integration of the wave equation, can be achieved by applying a time-domain bivariate pre-propagation filter to all source time functions followed by a time-domain bivariate post-propagation filter to the simulated field at all recording positions. The bivariate pre-propagation filter and the bivariate post-propagation filter are valid for any reasonable simulation time step. The two filters can be calculated once since they are independent of the simulation time step and they can be applied with any modeling scheme that uses second order time integration.

Finite-difference modeling of 2-D wave propagation in the vicinity of dipping interfaces: a comparison of anti-aliasing and equivalent medium approaches

“…A standard leap-frog time-stepping scheme will be used to step the solutions forward in time. Time-stepping errors are removed with time-dispersion transforms (Stork, 2013;Mittet, 2017Mittet, , 2019Koene et al, 2018;Wittsten et al, 2021). In practice, that entails pre-processing the source-time function with the forward time-dispersion transform, and post-processing recorded traces with the inverse time-dispersion transform.…”